Probability - the Australian Mathematical Sciences Institute

{36} • ProbabilityA1HA2A3A4A5A6εA partition of the event space and theintersection of an event H with the partition.The simplest version of a partition is any event A and its complement, since A ∩ A ′ = ∅and A ∪ A ′ = E .Note the event H represented on the diagram of the event space E above. It appears fromthe diagram that the probability of H can be obtained by summing the probabilities ofthe intersection of H with each event A i in the partition. We can show this formally:H = H ∩ E= H ∩ (A 1 ∪ A 2 ∪ ··· ∪ A k )= (H ∩ A 1 ) ∪ (H ∩ A 2 ) ∪ ··· ∪ (H ∩ A k ).Since A 1 , A 2 ,..., A k are mutually exclusive, it follows that the eventsH ∩ A 1 , H ∩ A 2 , ..., H ∩ A kare also mutually exclusive. Hence, by the third axiom of probability,Pr(H) = Pr(H ∩ A 1 ) + Pr(H ∩ A 2 ) + ··· + Pr(H ∩ A k )= Pr(A 1 )Pr(H|A 1 ) + Pr(A 2 )Pr(H|A 2 ) + ··· + Pr(A k )Pr(H|A k )k∑= Pr(A i )Pr(H|A i ).i=1This result is known as the law of total probability. Note that it does not matter if thereare some events A j in the partition for which H ∩ A j = ∅. (For example, see A 1 and A 6 inthe diagram above.) For these events, Pr(H ∩ A j ) = 0.A table can provide a useful alternative way to represent the partition and the event Hshown in the diagram above. In the following table, the event A 3 is used as an example.